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Characterizing proximity trees

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Abstract

Complete characterizations are given for those trees that can be drawn as either the relative neighborhood graph, relatively closest graph, Gabriel graph, or modified Gabriel graph of a set of points in the plane. The characterizations give rise to linear-time algorithms for determining whether a tree has such a drawing; if such a drawing exists one can be constructed in linear time in the real RAM model. The characterization of Gabriel graphs settles several conjectures of Matula and Sokal [17].

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References

  1. P. K. Agarwal and J. Matoušek. Relative Neighbourhood Graphs in Three Dimensions.Computational Geometry: Theory and Applications,2 (1992), 1–14.

    Google Scholar 

  2. J. A. Bondy and U. S. R. Murty.Graph Theory with Applications. Elsevier, New York, 1976.

    Google Scholar 

  3. R. J. Cimikowski. Properties of Some Euclidean Proximity Graphs.Pattern Recognition Letters,13 (1992), 417–423.

    Google Scholar 

  4. G. Di Battista, P. Eades, R. Tamassia, and I. G. Tollis. Algorithms for Automatic Graph Drawing: An Annotated Bibliography.Computational Geometry: Theory and Applications,4 (1994), 235–282.

    Google Scholar 

  5. G. Di Battista and L. Vismara. Angles of Planar Triangular Graphs.Proceedings of the ACM Symposium on Theory of Computing, 1993, pp. 431–437.

  6. M. B. Dillencourt. Graph-Theoretical Conditions for Inscribability and Delaunay Realizability. Technical Report 92-90, University of Califoria Irvine.

  7. P. Eades and S. Whitesides. Personal communication, 1993.

  8. H. Edelsbrunner, D. G. Kirkpatrick, and R. Seidel. On the Shape of a Set of Points in the Plane.IEEE Transaction on Information Theory,29 (1983), 551–559.

    Google Scholar 

  9. I. Fary. On Straight Lines Representation of Planar Graphs.Acta Scientiarum Mathematicarum (Szeged),11 (1948), 229–233.

    Google Scholar 

  10. K. R. Gabriel and R. R. Sokal. A New Statistical Approach to Geographical Analysis.Systematic Zoology,18 (1969), 54–64.

    Google Scholar 

  11. J. Garcia. A Counter-Example to a Theorem of Sendov. Manuscript in preparation.

  12. M. S. Jacobson, M. J. Lipman, and F. R. McMorris. Trees That Are Sphere-of-Influence Graphs. Technical Report 0191/2, University of Louisville.

  13. J. W. Jaromczyk and G. T. Toussaint. Relative Neighborhood Graphs and Their Relatives.Proceedings of the IEEE,80(9) (1992), 1502–1517.

    Google Scholar 

  14. D. G. Kirkpatrick and J. D. Radke. A Framework for Computational Morphology. InComputational Geometry, ed. G. T. Toussaint. Elsevier, Amsterdam, 1985, pp. 217–248.

    Google Scholar 

  15. P. M. Lankford. Regionalization: Theory and Alternative Algorithms.Geographical Analysis,1 (1969), 196–212.

    Google Scholar 

  16. A. Lubiw, and N. Sleumer, All Maximal Outerplanar Graphs Are Relative Neighborhood Graphs.Proceedings of the Fifth Canadian Conference on Computational Geometry, Waterloo, 1993, pp. 198–203.

  17. D. W. Matula and R. R. Sokal. Properties of Gabriel Graphs Relevant to Geographic Variation Research and the Clustering of Points in the Plane.Geographical Analysis,12(3) (1980), 205–222.

    Google Scholar 

  18. C. Monma and S. Suri. Transitions in Geometric Minimum Spanning Trees.Proceedings of the ACM Symposium on Computational Geometry, 1991, pp. 239–249.

  19. F. P. Preparata and M. I. Shamos,Computational Geometry —An Introduction. Springer-Verlag, New York, 1985.

    Google Scholar 

  20. J. D. Radke. On the Shape of a Set of Points. InComputational Morphology, ed. G. T. Toussaint. Elsevier, Amsterdam, 1988, pp. 105–136.

    Google Scholar 

  21. B. Sendov. Planar Neighbourhood Graphs Without Cycles.Comptes Rendus de l'Academie Bulgare des Sciences,44(4) (1991), 23–25.

    Google Scholar 

  22. M. I. Shamos and D. Hoey. Closest Point Problem.Proceedings of the 8th Annual IEEE Symposium on the Foundations of Computer Science, 1975, pp. 151–162.

  23. G. T. Toussaint. The Relative Neighborhood Graph of a Finite Planar Set.Pattern Recognition,12 (1980), 261–268.

    Google Scholar 

  24. G. T. Toussaint. Pattern Recognition and Geometrical Complexity.Proceedings of the Fifth International Conference on Pattern Recognition, Miami Beach, 1980, pp. 1340–1347.

  25. G. T. Toussaint. A Graph-Theoretical Primal Sketch. InComputational Morphology, ed. G. T. Toussaint. Elsevier, Amsterdam, 1988, pp. 229–260.

    Google Scholar 

  26. R. B. Urquhart. Some Properties of the Planar Euclidean Relative Neighbourhood Graph.Pattern Recognition Letters,1(5), 1983, 317–322.

    Google Scholar 

  27. R. C. Vetkamp. Theγ-Neighbourhood Graph.Computational Geometry: Theory and Applications,1 (1992), 227–246.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Département de Mathématiques et Informatique, Université du Québec à Trois-Rivières, C.P. 500, G9A 5H7, Trois-Rivières, Québec, Canada

    P. Bose

  2. Department of Computer Science, Williams College, 01267, Williamstown, MA, USA

    W. Lenhart

  3. Dipartimento di Informatica e Sistemistica, Università di Roma “La Sapienza”, via Salaria 113, I-00198, Roma, Italia

    G. Liotta

Authors
  1. P. Bose
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  2. W. Lenhart
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  3. G. Liotta
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Additional information

Communicated by G. Di Battista and R. Tamassia.

This research was conducted while the author was at the School of Computer Science of McGill University. Research supported in part by NSERC and FCAR.

This work was done when this author was visiting the School of Computer Science of McGill University.

This work was done when this author was visiting the School of Computer Science of McGill University.

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Bose, P., Lenhart, W. & Liotta, G. Characterizing proximity trees. Algorithmica 16, 83–110 (1996). https://doi.org/10.1007/BF02086609

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  • DOI: https://doi.org/10.1007/BF02086609

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